A Hierarchical Bayesian Model for Dating Species Divergences -- Evolution 2012
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This document describes a hierarchical Bayesian model for dating species divergences using fossil calibrations. It discusses: - Using fossil evidence to provide minimum age constraints for internal nodes - Specifying prior distributions on calibrated nodes that account for uncertainty in fossil ages - Implementing a Dirichlet process prior (DPP) as a hyperprior to model variation in rates of exponential calibration priors across multiple fossils - Testing the DPP hyperprior on simulated data to evaluate its ability to recover divergence times
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A Hierarchical Bayesian Model for Dating Species Divergences -- Evolution 2012
- 1. A B M D S D Tracy Heath Department of Integrative Biology, University of California, Berkeley Evolution 2012 Ottawa, Canada
- 2. D T E Goal: Estimate the ages of interior nodes Model how rates are Carettochelys Lissemys Apalone distributed across the tree Podocnemis Pelusios Pelomedusa Phrynops Chelus Elseya Chelodina Prior on node ages and Chelonia Dermochelys Chelydra external calibration Dermatemys Staurotypus Sternotherus information for estimates 95% CI Platysternon Geochelone Mauremys Heosemys of absolute node times No fossil Fossil calibration Emys Graptemys Trachemys P Triassic Jurassic Cretaceous Paleogene N 250 200 150 100 50 0 Time (My)
- 3. F C Fossil taxa are assigned to monophyletic clades Minimum age Time (My) Calibrating Divergence Times Fossil: Notogoneus osculus (Grande & Grande J. Paleont. 2008)
- 4. F C Age estimates from fossils can provide minimum time constraints for internal nodes Reliable maximum bounds are typically unavailable Minimum age Time (My) Calibrating Divergence Times
- 5. P D C N Parametric distributions are off-set by the age of the oldest fossil assigned Uniform (min, max) to a clade Log Normal (µ, σ2) These prior densities do not (necessarily) require Gamma (α, β) specification of maximum Exponential (λ) bounds Minimum age Time (My) Calibrating Divergence Times
- 6. P D C N Describe the waiting time between the divergence event and the age of the oldest fossil Minimum age Time (My) Calibrating Divergence Times
- 7. P D C N Overly informative priors can bias node age estimates to be too young Exponential (λ) Minimum age Time (My) Calibrating Divergence Times
- 8. P D C N Uncertainty in the age of the MRCA of the clade relative to the age of the fossil may be better captured by vague prior densities Exponential (λ) Minimum age Time (My) Calibrating Divergence Times
- 9. E D Prior density on calibrated nodes 60 Minimum age (fossil) Expected node age 80 λ = 5-1 100 Density 120 140 λ = 20-1 λ = 60-1 0 20 40 60 80 100 Node age - Fossil age Calibrating Divergence Times
- 10. P M C Precision of fossils as calibrations is affected by: • disparity in fossilization and preservation • geographical distribution • recovery bias • identification Calibrating Divergence Times
- 11. P M C It is unlikely that multiple fossil calibrations can be characterized by a single prior density Calibrating Divergence Times
- 12. P M C An appropriate prior for some nodes can also be an overly informative prior for other nodes Uncertainty in the time difference can be better captured by vague prior densities Calibrating Divergence Times
- 13. P M C Specifying appropriate prior densities for a range of minimum age constraints is a challenge for most molecular biologists Calibrating Divergence Times
- 14. P M C A hierarchical Bayesian framework can better reflect our statistical understanding of the distribution of ancestral node ages in relation to fossil calibrations Calibrating Divergence Times
- 15. A H B M From the bottom up: Hyperparameter Density The parameter χ is assumed to be drawn from an exponential Prior distribution Parameter Example: A Generic Hierarchical Bayesian Model
- 16. A H B M In Bayesian Hyperparameter inference, a Density parameter describing a prior distribution is called a hyperparameter Prior Parameter Example: A Generic Hierarchical Bayesian Model
- 17. A H B M Hyperparameter The exponential Density prior on χ has a hyperparameter: λ Prior Parameter Example: A Generic Hierarchical Bayesian Model
- 18. A H B M λ represents the rate of the exponential Hyperparameter distribution Density In a non-hierarchical model, the user is required to specify Prior the value of λ Parameter Example: A Generic Hierarchical Bayesian Model
- 19. A H B M Hyperprior: second order prior Density placed on a Hyperprior hyperparameter λ becomes a Density Hyperparameter random variable under the Prior hierarchical model Parameter Example: A Generic Hierarchical Bayesian Model
- 20. A H B M Hyperprior: Density Hyperprior values of χ are Density sampled by MCMC from a mixture of Hyperparameter exponential distributions Prior Parameter Example: A Generic Hierarchical Bayesian Model Markov chain Mote Carlo (MCMC)
- 21. A H B M Hyperprior: frees the user from the difficulty of Density Hyperprior specifying the value Density of λ Hyperparameter accounts for and quantifies uncertainty in the Prior hyperparameter Parameter Example: A Generic Hierarchical Bayesian Model
- 22. H C N Dirichlet process prior (DPP) on λ hyperparameters of exponential prior densities on multiple calibrated nodes Sample the time from 1 the MRCA to the fossil 2 from a mixture of 1 different exponential distributions 3 Account for uncertainty 2 in the rate of exponential calibration priors parameter classes DPP Hyperprior on Calibration-Node Prior Densities Heath, T.A. 2012 Syst. Biol. In press.
- 23. H C N The DPP models data as a mixture of distributions and can identify latent classes present in the data Calibration priors are 1 assumed to be clustered 2 into distinct λ-rate 1 parameter classes 3 (λ1 , λ2 , λ3 , . . . , λf ) ∼ DPP(α, G0 ) 2 f = number of fossil calibrations parameter classes DPP Hyperprior on Calibration-Node Prior Densities Heath, T.A. 2012 Syst. Biol. In press.
- 24. B I U DPP Current implementation: DPPDiv Availability: http://cteg.berkeley.edu/software.html • Divergence time estimation on a fixed topology Heath, Holder, Huelsenbeck. 2012. A Dirichlet process prior for estimating lineage-specific substitution rates Mol. Biol. Evol. 29:939–255. Heath. 2012. A hierarchical Bayesian model for calibrating estimates of species divergence times. Syst. Biol. (in press). Implementation Availability: DPPDiv @ http://cteg.berkeley.edu/software.html
- 25. H C N 1 The DPP hyperprior was 2 tested on simulated data, 1 using a model for 3 generating “fossil” calibrations for 100 2 simulation replicates parameter classes Evaluating the DPP Hyperprior with Simulated Data
- 26. T S Each model tree was generated under a constant-rate birth death process (20 extant taxa) 175 150 125 100 75 50 25 0 Time Modeling the Process of Fossilization
- 27. T S Each model tree was generated under a constant-rate birth death process (20 extant taxa) 175 150 125 100 75 50 25 0 Time Modeling the Process of Fossilization
- 28. F E Fossilization events were generated according to a Poisson process this example has 162 fossilization events 175 150 125 100 75 50 25 0 Time Modeling the Process of Fossilization
- 29. A S R Sampling Rate 0.2 1.05 The fossil sampling rate was evolved under an autocorrelated Brownian motion model 175 150 125 100 75 50 25 0 Time Modeling the Process of Preservation/Recovery
- 30. A S R Sampling Rate 0.2 1.05 The fossil sampling rate was evolved under an autocorrelated Brownian motion model 175 150 125 100 75 50 25 0 Time Modeling the Process of Preservation/Recovery
- 31. C S Sampling Rate 0.2 1.05 Recovered fossil 18 fossils were “recovered” in proportion to their sampling rates 175 150 125 100 75 50 25 0 Time Modeling the Process of Preservation/Recovery
- 32. R F The true phylogenetic placement of the recovered fossils is considered known 175 150 125 100 75 50 25 0 Time Modeling the Process of Preservation/Recovery
- 33. C F Only the oldest fossil assigned to a given node can be used for calibration 175 150 125 100 75 50 25 0 Time Modeling the Processes of Fossilization and Preservation/Recovery
- 34. C F Only the oldest fossil assigned to a given node can be used for calibration 175 150 125 100 75 50 25 0 Time Modeling the Processes of Fossilization and Preservation/Recovery
- 35. C F Only the oldest fossil assigned to a given node can be used for calibration 175 150 125 100 75 50 25 0 Time Modeling the Processes of Fossilization and Preservation/Recovery
- 36. C P: S 100 simulation replicates • Constant-rate birth-death trees; 20 taxa • Fossil calibrations generated under a Poisson-process model with sampling • Strict molecular clock 175 150 125 100 75 50 25 0 Time • Sequences generated under GTR+Γ Fossil Simulations: Data Generation
- 37. D T A Divergence time analyses of simulated data • Global/strict molecular clock • Birth-death tree prior • GTR+Γ • Fixed TRUE tree topology 175 150 125 100 75 50 25 0 Time Fossil Simulations: Analyses
- 38. A: F C P • Dirichlet process hyperprior (DPP-HP) • Fixed exponential distributions for each node • Informative prior (Fixed-λI ) • Vague prior (Fixed-λV ) 175 150 125 100 Time 75 50 25 0 • True prior (Fixed-λT ) Fossil Simulations: Analyses
- 39. F C P A fixed exponential prior distribution on each calibrated node calibration difference Prior Density 25 30 40 45 50 55 60 True Fossil Node Age Fossil Simulations: Analyses Priors: Fixed-λV , Fixed-λI , Fixed-λT
- 40. F C P A fixed exponential prior distribution on each calibrated node calibration difference Prior Density 25 30 40 45 50 55 60 True Fossil Node Age Fossil Simulations: Analyses Priors: Fixed-λV , Fixed-λI , Fixed-λT
- 41. E P Density Node Age 95% Credible Interval (CI): 95% CI True Age A measure of uncertainty Density Approximation of the interval containing 95% of the highest Node Age posterior density Density Node Age Simulations: Methods
- 42. E P Density Node Age 95% CI Coverage Probability: True Age Density The proportion of the time the 95% CI contains the true Node Age value is a measure of accuracy Density Node Age Simulations: Methods
- 43. N A: C P For each calibration prior: the proportion of nodes (across all simulation replicates) where the TRUE node age was contained within the 95% CI Calibration Coverage Prior probability DPP-HP 0.926 Fixed-λT 0.917 Fixed-λV 0.843 Fixed-λI 0.550 Fossil Simulations: Results Priors: Fixed-λV , Fixed-λI , Fixed-λT , DPP-HP
- 44. N A: C P 1 0.8 Coverage probability 0.6 0.4 Fixed-λV (Vague) 0.2 Fixed-λI (Inform.) Fixed-λT (True) DPP-HP 0 0.1 1 10 100 1000 True Node Age (log scale) Fossil Simulations: Results Priors: Fixed-λV , Fixed-λI , Fixed-λT , DPP-HP
- 45. T N C Accuracy Coverage proability: 1 As the number of 0.8 Coverage Probability fossils increases from 5 to 11, the 0.6 coverage probability stays relatively high 0.4 for Fixed-λT and 0.2 DPP-HP Fixed-λV (Vague) Fixed-λI (Inform.) Fixed-λT (True) DPP-HP 0 4 5 6 7 8 9 10 11 12 Number of Calibrating Fossils Fossil Simulations: Results Priors: Fixed-λV , Fixed-λI , Fixed-λT , DPP-HP
- 46. T N C Accuracy 1 Coverage proability: 0.8 Coverage Probability Under Fixed-λI , adding fossils with 0.6 overly informative priors decreases 0.4 coverage probability 0.2 Fixed-λV (Vague) Fixed-λI (Inform.) Fixed-λT (True) DPP-HP 0 4 5 6 7 8 9 10 11 12 Number of Calibrating Fossils Fossil Simulations: Results Priors: Fixed-λV , Fixed-λI , Fixed-λT , DPP-HP
- 47. N A: C P Node age estimates under the DPP Hyperprior are, on average, the same as when the expectations of calibration densities are fixed to the TRUE node age Calibration using 1 DPP Hyperprior 0.8 Coverage probability does not require 0.6 specification of the calibration density 0.4 hyperparameters 0.2 Fixed-λV (Vague) Fixed-λI (Inform.) Fixed-λT (True) DPP-HP 0 0.1 1 10 100 1000 True Node Age (log scale) Fossil Simulations: Results Priors: Fixed-λV , Fixed-λI , Fixed-λT , DPP-HP
- 48. C N A E 175 150 125 100 75 50 25 0 Time Fossil Simulations: Results
- 49. C N A E Root Node (R) Node A Node B 0.1 0.7 0.35 0.09 Fixed-λV 0.6 0.3 0.08 Fixed-λI 0.07 0.5 Fixed-λT 0.25 Density 0.06 0.4 DPP-HP 0.2 0.05 0.04 0.3 0.15 0.03 0.2 0.1 0.02 0.1 0.05 0.01 0 0 0 160 180 200 135 140 145 150 30 40 50 Fossil True Fossil True Fossil True Node C Node D Node E 0.12 0.6 0.3 0.1 0.5 0.25 Density 0.08 0.4 0.2 0.06 0.3 0.15 0.04 0.2 0.1 0.02 0.1 0.05 0 0 0 60 80 100 120 10 20 30 40 110 120 130 140 150 Fossil True Fossil True Fossil True Node Age Fossil Simulations: Results Priors: Fixed-λV , Fixed-λI , Fixed-λT , DPP-HP
- 50. H C N Dirichlet process prior on hyperparameters of prior densities on multiple calibrated nodes allows for flexible integration of data from the fossil record 1 2 Node ages are 1 sampled from a mixture of 3 exponential distributions 2 parameter classes Conclusions Availability: DPPDiv @ http://cteg.berkeley.edu/software.html
- 51. H C N Dirichlet process prior on hyperparameters of prior densities on multiple calibrated nodes allows for flexible integration of data from the fossil record 1 Identify fossils that 2 may share similar 1 properties Free user from 3 responsibility of 2 specifying hyperparameters parameter classes Conclusions Availability: DPPDiv @ http://cteg.berkeley.edu/software.html
- 52. C M M Sampling Rate 0.2 1.05 Modeling branching patterns AND fossilization, preservation, and recovery for use as priors for divergence time estimation 175 150 125 100 75 50 25 0 Time Models of Fossilization and Preservation for Bayesian Inference
- 53. C M M Sampling Rate 0.2 1.05 Incorporate more information from the fossil and rock records and construct better and more realistic tree priors 175 150 125 100 75 50 25 0 Time Models of Fossilization and Preservation for Bayesian Inference
- 54. A Funding: • NSF Postdoc Fellowship DBI-0805631 • NIH grants GM-069801 & GM-086887 Thanks to: • Mark Holder, John Huelsenbeck, Brian Moore, Tom Near DPPDiv: http://cteg.berkeley.edu/software.html Slides available at: http://www.slideshare.net/trayc7 Thanks!